Quantum channels preserving sigma-additivity and Ulam measurable cardinals

This paper investigates the interplay between the properties of quantum states on the Hilbert space ell₂(κ) and the set-theoretic nature of the cardinal κ. We focus on the existence of singular σ-additive states - functionals whose induced measures are σ-additive yet vanish on singletons. While the existence of such states is known to be equivalent to the Ulam measurability of κ, their structural and dynamical properties remain largely unexplored. We prove that any σ-additive state on the diagonal algebra is representable as a Pettis integral over a singular σ-additive measure, extending the classical representation theory to the non-normal sector. Furthermore, we construct a class of quantum channels using σ-complete ultrafilters that map normal states to singular σ-additive states, effectively <<archiving>> information into the singular part of the state space.